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Optimal control problems for the bilinear system of special structure

V. A. Srochko^a, E. V. Aksenyushkina<sup>b

a</sup> Irkutsk State University, 1, K. Marx st., Irkutsk, 664003
^b Baikal State University, 11, Lenin st., Irkutsk, 664015

Abstract: We consider three optimal control problems (linear terminal, bilinear and quadratic functionals) with respect to the special bilinear system with a matrix of rank 1. For the terminal problem we received two versions of conditions on the initial data of the system and functional in which the maximum principle becomes the sufficient optimality condition. At the same time the problem becomes very simple: the optimal control is determined in the process of integration phase or conjugate system (one Cauchy problem).
Next the problem of optimization of bilinear functional is considered. Sufficient optimality conditions for the boundary controls without switching points are obtained.These conditions are represented as inequalities for functions of one variable (the time).
The optimal control problem with the quadratic functional reduces to bilinear case on the basis of special increment formula.

Keywords: bilinear system, optimal control problem, the maximum principle, sufficient optimality conditions.

UDC: 517.97

MSC: 49J15